addition

Addition is associative – that is, a series of additions can be carried out in any order without affecting the result. This diagram shows the effects of successively weighing [A] 3, 4, and then 5 units of a substance on a spring balance and [B] weighing 4, 5, and then 3 units. In both cases the total weight – the sum or the additions – is 2 units. As in many other mathematical laws, this is applied common sense.

Addition is one of the four fundamental operations
of arithmetic. It is denoted by the sign "+".

The numbers which are added are called summands. For example,
4 (a summand) + 3 (another summand) = 7 (sum).

Addition of fractions

Fractions can be added directly only if
they have equal denominators, e.g., 3/17 + 4/17 = 7/17.

If fractions with unequal denominators are to be added they must first be
brought to their least common denominator.

Addition of literal numbers

Identically-named numbers can be added (and subtracted) by adding the coefficients:

Addition of powers and surds

Powers and surds must be treated like literal
numbers for the purposes of addition, e.g.,

a2 + 4b3 + 4c3 +4a2 = 5a2 + 4(b3 + c3;
3√2 + 5√3 + √2 = 4√2 + 5√3

Addition of complex numbers

In adding complex numbers the real
and imaginary parts must be added separately, e.g.,

(5 + 3i) + (17 - i) = 22 + 2i.

Addition in terms of sets

The addition of positive numbers can best be defined in terms of set
theory; if one considers set A to contain 4 elements, set B to contain 5 elements, then A U B (see union)
contains 9 elements; i.e., 4 + 5 = 9. The addition of negative numbers is
equivalent to subtraction in that a + (-b) is equivalent
to a - b.